code: def test_inverse_laplace_transform_old():
      from sympy.functions.special.delta_functions import DiracDelta
  
      ILT = inverse_laplace_transform
      a, b, c, d = symbols("a b c d", positive=True)
      n, r = symbols("n, r", real=True)
      t, z = symbols("t z")
      f = Function("f")
      F = Function("F")
  
      def simp_hyp(expr):
          return factor_terms(expand_mul(expr)).rewrite(sin)
  
      L = ILT(F(s), s, t)
      assert laplace_correspondence(L, {f: F}) == f(t)
      assert ILT(exp(-a * s) / s, s, t) == Heaviside(-a + t)
      assert ILT(exp(-a * s) / (b + s), s, t) == exp(
          -b * (-a + t)
      ) * Heaviside(-a + t)
      assert ILT((b + s) / (a**2 + (b + s) ** 2), s, t) == exp(
          -b * t
      ) * cos(a * t) * Heaviside(t)
      assert ILT(exp(-a * s) / s**b, s, t) == (-a + t) ** (
          b - 1
      ) * Heaviside(-a + t) / gamma(b)
      assert ILT(exp(-a * s) / sqrt(s**2 + 1), s, t) == Heaviside(
          -a + t
      ) * besselj(0, a - t)
      assert ILT(1 / (s * sqrt(s + 1)), s, t) == Heaviside(t) * erf(
          sqrt(t)
      )
      # TODO sinh/cosh shifted come out a mess. also delayed trig is a mess
      # TODO should this simplify further?
      assert ILT(exp(-a * s) / s**b, s, t) == (t - a) ** (
          b - 1
      ) * Heaviside(t - a) / gamma(b)
      assert ILT(exp(-a * s) / sqrt(1 + s**2), s, t) == Heaviside(
          t - a
      ) * besselj(
          0, a - t
      )  # note: besselj(0, x) is even
      # XXX ILT turns these branch factor into trig functions ...
      assert simplify(
          ILT(
              a**b
              * (s + sqrt(s**2 - a**2)) ** (-b)
              / sqrt(s**2 - a**2),
              s,
              t,
          ).rewrite(exp)
      ) == Heaviside(t) * besseli(b, a * t)
      assert ILT(
          a**b
          * (s + sqrt(s**2 + a**2)) ** (-b)
          / sqrt(s**2 + a**2),
          s,
          t,
          simplify=True,
      ).rewrite(exp) == Heaviside(t) * besselj(b, a * t)
      assert ILT(1 / (s * sqrt(s + 1)), s, t) == Heaviside(t) * erf(
          sqrt(t)
      )
      # TODO can we make erf(t) work?
      assert ILT(
          (s * eye(2) - Matrix([[1, 0], [0, 2]])).inv(), s, t
      ) == Matrix(
          [[exp(t) * Heaviside(t), 0], [0, exp(2 * t) * Heaviside(t)]]
      )
      # Test time_diff rule
      assert ILT(s**42 * f(s), s, t) == Derivative(
          InverseLaplaceTransform(f(s), s, t, None), (t, 42)
      )
      assert ILT(cos(s), s, t) == InverseLaplaceTransform(
          cos(s), s, t, None
      )
      # Rules for testing different DiracDelta cases
      assert ILT(
          2 * exp(3 * s) - 5 * exp(-7 * s), s, t
      ) == 2 * InverseLaplaceTransform(
          exp(3 * s), s, t, None
      ) - 5 * DiracDelta(
          t - 7
      )
      a = cos(sin(7) / 2)
      assert ILT(a * exp(-3 * s), s, t) == a * DiracDelta(t - 3)
      assert ILT(exp(2 * s), s, t) == InverseLaplaceTransform(
          exp(2 * s), s, t, None
      )
      r = Symbol("r", real=True)
      assert ILT(exp(r * s), s, t) == InverseLaplaceTransform(
          exp(r * s), s, t, None
      )
      # Rules for testing whether Heaviside(t) is treated properly in diff rule
      assert ILT(s**2 / (a**2 + s**2), s, t) == (
          -a * sin(a * t) * Heaviside(t) + DiracDelta(t)
      )
      assert ILT(s**2 * (f(s) + 1 / (a**2 + s**2)), s, t) == (
          -a * sin(a * t) * Heaviside(t)
          + DiracDelta(t)
          + Derivative(
              InverseLaplaceTransform(f(s), s, t, None), (t, 2)
          )
      )
      # Rules from the previous test_inverse_laplace_transform_delta_cond():
      assert ILT(exp(r * s), s, t, noconds=False) == (
          InverseLaplaceTransform(exp(r * s), s, t, None),
          True,
      )
      # inversion does not exist: verify it doesn't evaluate to DiracDelta
      for z in (
          Symbol("z", extended_real=False),
          Symbol("z", imaginary=True, zero=False),
      ):
          f = ILT(exp(z * s), s, t, noconds=False)
          f = f[0] if isinstance(f, tuple) else f
          assert f.func != DiracDelta
  
imports: from sympy.integrals.laplace import laplace_transform, inverse_laplace_transform, LaplaceTransform, InverseLaplaceTransform, _laplace_deep_collect, laplace_correspondence, laplace_initial_conds
  from sympy.core.function import Function, expand_mul
  from sympy.core import EulerGamma, Subs, Derivative, diff
  from sympy.core.exprtools import factor_terms
  from sympy.core.numbers import I, oo, pi
  from sympy.core.relational import Eq
  from sympy.core.singleton import S
  from sympy.core.symbol import Symbol, symbols
  from sympy.simplify.simplify import simplify
  from sympy.functions.elementary.complexes import Abs, re
  from sympy.functions.elementary.exponential import exp, log, exp_polar
  from sympy.functions.elementary.hyperbolic import cosh, sinh, coth, asinh
  from sympy.functions.elementary.miscellaneous import sqrt
  from sympy.functions.elementary.piecewise import Piecewise
  from sympy.functions.elementary.trigonometric import atan, cos, sin
  from sympy.logic.boolalg import And
  from sympy.functions.special.gamma_functions import lowergamma, gamma, uppergamma
  from sympy.functions.special.delta_functions import DiracDelta, Heaviside
  from sympy.functions.special.singularity_functions import SingularityFunction
  from sympy.functions.special.zeta_functions import lerchphi
  from sympy.functions.special.error_functions import fresnelc, fresnels, erf, erfc, Ei, Ci, expint, E1
  from sympy.functions.special.bessel import besseli, besselj, besselk, bessely
  from sympy.testing.pytest import slow, warns_deprecated_sympy
  from sympy.matrices import Matrix, eye
  from sympy.abc import s
label_desc: Use image node_id 3 for calling a global function with example usage: test_inverse_laplace_transform_old() and returns: factor_terms